• With your partner, discuss and draft a definition for a spiral. Share your definition with the class, and formulate a class definition.

• The first Baravelle spiral we will construct is formed by a group of nested equilateral triangles. Let's begin by constructing a large equilateral triangle, T₀ , in your sketch window. Construct the midpoints of each of the three sides of the triangle and connect them by segments. The figure you have constructed separates your equilateral triangle into smaller triangles. How many smaller triangles are there? What kind of triangles are they? What other conjectures can you make about the four triangles? (Instructor Note: A Sketchpad file illustrating the construction of an equilateral triangle has been saved as equil.gsp.)
  
  
• Construct the polygon interior of one of the corner triangles and label it T₁. Formally prove that T₁ is congruent to the three other inner triangles. The area of T₁ is what fraction of T₀? Verify your conjecture by a formal proof.

To label a figure in a sketch window: Select the Text Tool   in the tool box. Double-click anywhere in your sketch window. When a flashing cursor appears, you may type in the box. To edit your text box once you’ve clicked outside of it, select the Text Tool and click inside the text box. A flashing cursor should appear, allowing you to edit.

• We would like to continue the construction of the Baravelle spiral by finding and connecting the midpoints of each of the sides of the center triangle. The figure you have constructed separates the center equilateral triangle into smaller triangles. How many smaller triangles are there? What kind of triangles are they? What else can you say about the four smaller triangles?

• Moving in a clockwise manner, construct the polygon interior of the corner inner triangle, which is adjacent to T₁, and label it T₂ . Be sure to keep the color of your polygon interiors consistent. The area of T₂ is what fraction of T₀?

• We have created two levels of this construction. Create two or three additional shaded triangles using the same process as above. If this construction was continued indefinitely, the resulting shaded region is called a Baravelle spiral. 

Part 2:

• Looking at your constructed Baravelle spiral, visually estimate the spiral’s area as a fraction of the area of T₀. Justify your answer. How could we find the actual area of the spiral?
  
  
• In your own words, what is a series? How does a series differ from a sequence? Can an infinite series sum to a finite number? If so, how can we find that number?
  
  
• Record the ratios of the areas of the shaded triangles, T_N , in comparison to the area of the original equilateral triangle, T₀. Fill in the chart below, computing the partial sums of the areas of each of the levels of triangles.

• What did you notice about consecutive ratios of the areas of the triangles? Furthermore, what did you notice about the column of computed partial sums? As N increased, what number did the column of partial sums approach? Describe your observations in your own words. Use your knowledge of infinite series to verify this sum. See the spreadsheet activity Exploring and Analyzing Sequences.

• Express the area of the Baravelle spiral as an infinite series. The spiral's area is what fraction of the area of the original triangle?

• Shade (each in a different color) two other Baravelle spirals using your original equilateral triangle. (Instructor Note: A Sketchpad file illustrating the construction of a Baravelle spiral generated by an equilateral triangle has been saved as eqtri-spiral.gsp.)

• Another infinite series is similarly found by considering the length of the outer edge of the spiral. The outer edge of a spiral is the sum of the lengths of one side of each of the shaded triangles. Record the length of each side of the triangle, T_N , in comparison to the length of each side of the original equilateral triangle, T₀. Fill in the chart below, computing the partial sums of the length of the outer edge for each of the levels of triangles.

• What did you notice about consecutive ratios of the lengths of a side of the triangles? Furthermore, what did you notice about the column of computed partial sums? As N increased, what number did the column of partial sums approach? Describe your observations in your own words. Use your knowledge of infinite series to verify this sum.
  
  
• Express the outer edge of the spiral as an infinite series.

Part 4:

"Mathematically, iteration refers to the process of repeatedly applying some mathematical construction, calculation, or other operation to the previous result of that same operation. The operation must define an output in terms of some input, and the iteration uses the output of one step as the input for the next step" (Key Press, 2001).

• As you can see from the construction in Part 1, the process to construct a Baravelle spiral is a iterative process. In this context, discuss with your neighbor what it means to be a iterative process? Share your ideas with the class.

• To further explore infinite series and iterations, we will direct focus on a Baravelle spiral generated by a square. To begin, construct a large square in your sketch window. (Instructor Note: A Sketchpad file illustrating the construction of a square has been saved as square.gsp.)

The basic construction for a Baravelle spiral is to find and connect the midpoints of each of the sides of the center square multiple times. Sketchpad allows us to create an iterated construction of one or more objects in terms of points and parameters.  Follow the steps in the gray box below to create an iterative construction of a Baravelle spiral generated by a square.

• The figure you have constructed separates the original square into a smaller squares and triangles. Within the first level of the construction, how many smaller triangles are there? What kind of triangles are they? What other conjectures can you state about the four corner triangles? 

• Record the ratios of the areas of the shaded triangles, T_N , in comparison to the area of the original square. Fill in the chart below, computing the partial sums of the areas of each of the levels of triangles.

• What did you notice about consecutive ratios of the areas of the triangles? Furthermore, what did you notice about the column of computed partial sums? As N increased, what did the column of partial sums approach?

• Express the area of the spiral as an infinite series. The spiral’s area is what fraction of the area of the original square?

• Construct the three other Baravelle spirals using your original square, each in a different color. (Instructor Note: A Sketchpad file illustrating the construction of a Baravelle spiral generated by a square has been saved as square-spiral.gsp.)

• Explore a similar investigation using a hexagon as the original starting figure. By constructing the midpoints of each of the six sides of the hexagon and connecting them by segments, continue a similar procedure as the triangle and square. The shaded spiral's area is approximately what fraction of the original hexagon? Use your knowledge about infinite series to verify this sum?
  
  
• In each of the Baravelle spirals constructed we constructed the midpoints of each of the sides of our polygon and connected them by segments. A different type of spiral could be constructed by choosing points other than the midpoints of each of the sides of the polygon. Iteratively construct a spiral that does not use the midpoints of the sides of the polygon. Write a report describing your construction and showing examples of your spirals.